Some fixed point theorems on non-convex sets

M. Radhakrishnan, S. Rajesh, Sushama Agrawal
DOI: https://doi.org/10.4995/agt.2017.7452

Abstract

In this paper, we prove that if $K$ is a nonempty weakly compact set in a Banach space $X$, $T:K\to K$ is a nonexpansive map satisfying $\frac{x+Tx}{2}\in K$ for all $x\in K$ and if $X$ is $3-$uniformly convex or $X$ has the Opial property, then $T$ has a fixed point in $K.$

Keywords

fixed points; nonexpansive mappings; T-regular set; k-uniform convex Banach spaces; Opial property

Subject classification

47H09; 47H10

Full Text:

PDF

References

J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, in Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 1997.

F. E. Browder, Nonexpansive nonlinear operations in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041-1043.

https://doi.org/10.1073/pnas.54.4.1041

M. Edelstein, The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc. 78 (1972), 206-208.

https://doi.org/10.1090/S0002-9904-1972-12918-5

K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math. 21 (1983), 115-123. https://doi.org/10.1090/conm/021/729507

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., Cambridge Univ. Press, 1990.

https://doi.org/10.1017/CBO9780511526152

K. Goebel and R. Schöneberg, Moons, bridges, birds and nonexpansive mappings in Hilbert space, Bull. Austral. Math. Soc. 17 (1977), 463-466.

https://doi.org/10.1017/S0004972700010728

J. P. Gossez and L. Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 563-573.

https://doi.org/10.2140/pjm.1972.40.565

D. Göhde, Zum Prinzip der knontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258.

https://doi.org/10.1002/mana.19650300312

A. R. Khan and N. Hussain, Iterative approximation of fixed points of nonexpansive maps, Sci. Math. Jpn. 4 (2001), 749-757.

W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72(1965), 1004-1006.

https://doi.org/10.2307/2313345

W. A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and $k$-uniform rotundity, Proc. Sympos. Pure Math. 45 (1986), 51-64.

https://doi.org/10.1090/pspum/045.2/843594

M. A. Krasnoselskii, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.) 10 (1955), 123-–127.

T. C. Lim, On moduli of $k-$convexity, Abstr. Appl. Anal. 4 (1999), 243-247.

https://doi.org/10.1155/S1085337599000202

P. K. Lin, $k-$uniform rotundity is equivalent to $k-$uniform convexity, J. Math. Anal. Appl. 132 (1988), 349-355.

https://doi.org/10.1016/0022-247X(88)90066-2

P. K. Lin, K. K. Tan and H. K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Analysis, Theory, Methods and Applications 24 (1995), 929-946.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

https://doi.org/10.1090/S0002-9904-1967-11761-0

E. Sliverman, Definitions of Lebesgue area for surfaces in metric spaces, Rivista Mat. Univ. Parma 2 (1951), 47-76.

F. Sullivan, A generalization of uniformly rotund Banach spaces, Canad. J. Math. 31 (1979), 628-636.

https://doi.org/10.4153/CJM-1979-063-9

T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008), 1088-1095.

https://doi.org/10.1016/j.jmaa.2007.09.023

P. Veeramani, On some fixed point theorems on uniformly convex Banach spaces, J. Math. Anal. Appl. 167 (1992), 160-166.

https://doi.org/10.1016/0022-247X(92)90243-7

Abstract Views

2802
Metrics Loading ...

Metrics powered by PLOS ALM


Bookmark and Share


Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt